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Properties

Label 10-6027e5-1.1-c1e5-0-0
Degree $10$
Conductor $7.953\times 10^{18}$
Sign $1$
Analytic cond. $2.58161\times 10^{8}$
Root an. cond. $6.93727$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 5·3-s + 5·4-s + 9·5-s + 15·6-s + 7·8-s + 15·9-s + 27·10-s + 25·12-s + 3·13-s + 45·15-s + 10·16-s + 16·17-s + 45·18-s − 4·19-s + 45·20-s − 3·23-s + 35·24-s + 38·25-s + 9·26-s + 35·27-s + 13·29-s + 135·30-s + 4·31-s + 14·32-s + 48·34-s + 75·36-s + ⋯
L(s)  = 1  + 2.12·2-s + 2.88·3-s + 5/2·4-s + 4.02·5-s + 6.12·6-s + 2.47·8-s + 5·9-s + 8.53·10-s + 7.21·12-s + 0.832·13-s + 11.6·15-s + 5/2·16-s + 3.88·17-s + 10.6·18-s − 0.917·19-s + 10.0·20-s − 0.625·23-s + 7.14·24-s + 38/5·25-s + 1.76·26-s + 6.73·27-s + 2.41·29-s + 24.6·30-s + 0.718·31-s + 2.47·32-s + 8.23·34-s + 25/2·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 7^{10} \cdot 41^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 7^{10} \cdot 41^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(3^{5} \cdot 7^{10} \cdot 41^{5}\)
Sign: $1$
Analytic conductor: \(2.58161\times 10^{8}\)
Root analytic conductor: \(6.93727\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 3^{5} \cdot 7^{10} \cdot 41^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1059.744587\)
\(L(\frac12)\) \(\approx\) \(1059.744587\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$ \( ( 1 - T )^{5} \)
7 \( 1 \)
41$C_1$ \( ( 1 - T )^{5} \)
good2$C_2 \wr S_5$ \( 1 - 3 T + p^{2} T^{2} - p^{2} T^{3} + 3 T^{4} - T^{5} + 3 p T^{6} - p^{4} T^{7} + p^{5} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
5$C_2 \wr S_5$ \( 1 - 9 T + 43 T^{2} - 138 T^{3} + 349 T^{4} - 791 T^{5} + 349 p T^{6} - 138 p^{2} T^{7} + 43 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 31 T^{2} - 8 T^{3} + 530 T^{4} - 64 T^{5} + 530 p T^{6} - 8 p^{2} T^{7} + 31 p^{3} T^{8} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 3 T + 9 T^{2} + 36 T^{3} - 21 T^{4} + 269 T^{5} - 21 p T^{6} + 36 p^{2} T^{7} + 9 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 16 T + 145 T^{2} - 940 T^{3} + 4958 T^{4} - 21996 T^{5} + 4958 p T^{6} - 940 p^{2} T^{7} + 145 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 4 T + 77 T^{2} + 272 T^{3} + 2632 T^{4} + 7516 T^{5} + 2632 p T^{6} + 272 p^{2} T^{7} + 77 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 + 3 T + 61 T^{2} + 96 T^{3} + 2167 T^{4} + 3263 T^{5} + 2167 p T^{6} + 96 p^{2} T^{7} + 61 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 13 T + 169 T^{2} - 1356 T^{3} + 10309 T^{4} - 57051 T^{5} + 10309 p T^{6} - 1356 p^{2} T^{7} + 169 p^{3} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 4 T + 105 T^{2} - 424 T^{3} + 5688 T^{4} - 17740 T^{5} + 5688 p T^{6} - 424 p^{2} T^{7} + 105 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 17 T + 195 T^{2} - 1414 T^{3} + 8865 T^{4} - 50063 T^{5} + 8865 p T^{6} - 1414 p^{2} T^{7} + 195 p^{3} T^{8} - 17 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 6 T + 5 p T^{2} - 1020 T^{3} + 10 p^{2} T^{4} - 65536 T^{5} + 10 p^{3} T^{6} - 1020 p^{2} T^{7} + 5 p^{4} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 15 T + 135 T^{2} - 856 T^{3} + 3013 T^{4} - 10743 T^{5} + 3013 p T^{6} - 856 p^{2} T^{7} + 135 p^{3} T^{8} - 15 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 11 T + 125 T^{2} - 580 T^{3} + 3303 T^{4} - 8879 T^{5} + 3303 p T^{6} - 580 p^{2} T^{7} + 125 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 12 T + 219 T^{2} + 2328 T^{3} + 23990 T^{4} + 189276 T^{5} + 23990 p T^{6} + 2328 p^{2} T^{7} + 219 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 12 T + 87 T^{2} + 400 T^{3} + 8096 T^{4} + 79700 T^{5} + 8096 p T^{6} + 400 p^{2} T^{7} + 87 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 11 T + 103 T^{2} - 472 T^{3} + 4533 T^{4} - 28787 T^{5} + 4533 p T^{6} - 472 p^{2} T^{7} + 103 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 18 T + 337 T^{2} - 4140 T^{3} + 44308 T^{4} - 405432 T^{5} + 44308 p T^{6} - 4140 p^{2} T^{7} + 337 p^{3} T^{8} - 18 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 12 T + 331 T^{2} + 2748 T^{3} + 43140 T^{4} + 270356 T^{5} + 43140 p T^{6} + 2748 p^{2} T^{7} + 331 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 23 T + 511 T^{2} - 7136 T^{3} + 89153 T^{4} - 841307 T^{5} + 89153 p T^{6} - 7136 p^{2} T^{7} + 511 p^{3} T^{8} - 23 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 2 T + 151 T^{2} + 8 p T^{3} + 9314 T^{4} + 129172 T^{5} + 9314 p T^{6} + 8 p^{3} T^{7} + 151 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 28 T + 625 T^{2} - 9864 T^{3} + 127142 T^{4} - 1312264 T^{5} + 127142 p T^{6} - 9864 p^{2} T^{7} + 625 p^{3} T^{8} - 28 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 3 T + 409 T^{2} + 1104 T^{3} + 72249 T^{4} + 157985 T^{5} + 72249 p T^{6} + 1104 p^{2} T^{7} + 409 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.75295589901361320533663089592, −4.51328849396837791024970531098, −4.34125898800702687694513511858, −4.27345397014890054531746977186, −4.23734686382537999613059632010, −3.77165260629676126357295675583, −3.75825777288521191574581544760, −3.56509586876517393549603126118, −3.52073874747863369292634900805, −3.33811784159214735631397271985, −3.16471821068399939410355945278, −2.75991820994189816264611840850, −2.74643425989658244136702691761, −2.63303086744109500513969440545, −2.40724763579533302503369882199, −2.34634230079349469440253768781, −2.32219096709574346265436308381, −2.20079564898418500830925460678, −1.83181431589978791015820321121, −1.41926032489317530332869323248, −1.36182510387834586760067752943, −1.25026646332931254007694346782, −1.17942930154216412667295704251, −0.900159543217372559090614680602, −0.72996753407250586472582182513, 0.72996753407250586472582182513, 0.900159543217372559090614680602, 1.17942930154216412667295704251, 1.25026646332931254007694346782, 1.36182510387834586760067752943, 1.41926032489317530332869323248, 1.83181431589978791015820321121, 2.20079564898418500830925460678, 2.32219096709574346265436308381, 2.34634230079349469440253768781, 2.40724763579533302503369882199, 2.63303086744109500513969440545, 2.74643425989658244136702691761, 2.75991820994189816264611840850, 3.16471821068399939410355945278, 3.33811784159214735631397271985, 3.52073874747863369292634900805, 3.56509586876517393549603126118, 3.75825777288521191574581544760, 3.77165260629676126357295675583, 4.23734686382537999613059632010, 4.27345397014890054531746977186, 4.34125898800702687694513511858, 4.51328849396837791024970531098, 4.75295589901361320533663089592

Graph of the $Z$-function along the critical line